Investigation of chaotic pattern of discharge in Dez River under different time scales

River flow investigation is one of the main issues in hydrology and water resources studies. Chaos analyses can be used to analyze the river flow time series as a dynamic system with highly sensitive to initial conditions. The chaos theory, which is based on non-linear dynamic systems, has resulted in a great change in understanding and expressing different phenomena. This theory deals with the study of systems that at first glance may seem irregular; but in fact they are governed by clear rules. Such systems are very sensitive to initial conditions, such that minor inputs could have significant impacts. Since investigating the presence of different characteristics at different time-scales in rivers is one of the main challenges of hydrology, the aim of this paper was to study the behavior of river flow process at different time scales. Despite chaotic studies conducted on river-flow discharges, these analysis of Dez River discharge, located in southwest of Iran, has not been implemented for different time scales. In this study, based on the flow-discharge data of Bamdezh station on Dez River, the presence of chaos in daily, monthly and seasonal scales is discussed.
Dez River consists of two main branches, namely Sezar and Bakhtiari, and after exiting from the mountainous region in the north of Andimeshk and Dezful cities, enters into the Khuzestan plain. After passing the city of Dezful, and running through a 186 km meandering path (from Dezful to Band-e Ghir), it joins Shotait and Gargar Rivers in Band-e Ghir and forms Great Karun River, which flows toward city of Ahvaz. Bamdezh is the last hydrometric station on Dez River, before joining the great Karun River. Daily and monthly discharge data of Bamdezh station during 31 water-years (1981 to 2011) were used. Four non-linear dynamic fallowing methods were used: 1) phase space reconstruction, 2) correlation dimension method, 3) determination the greatest Lyapunov exponent and 4) calculation the Hurst exponent. The state (phase) space is a useful tool to study the dynamic systems. According to this concept, a dynamic system can be described by means of a state space diagram. Each dynamic system consisted of differential equations with partial derivatives. To determine these equations and their type, the embedding dimension and delay time parameters have to be determined. The delay time could be obtained from the method of assessment of correlation function (ACF) or average mutual information (AMI). In this study, the average mutual information method was used to estimate delay time. In this method, time of the occurrence of first minimum in the average mutual information function is selected as the appropriate delay time. The embedding dimension is obtained from the false nearest neighbor (FNN) method.
The obtained results showed that delay time for daily, monthly and seasonal time-scales is 80 days, 2 months and 2 seasons, respectively, and optimal embedding dimension is 10, 3 and 1, respectively. The correlation dimension was calculated to determine the chaotic nature of the system. Results revealed that correlation dimension at daily and monthly scales, due to saturation of the diagram, was 3.765 and 3.84, respectively. Therefore, Dez River system is chaotic in these two time scales. But at the seasonal time scale, the diagram trend was ascending and as a result the river discharge is random. Another indicative criterion of the chaotic system is the greatest Lyapunov exponent. By using this exponent, the behavior could be determined in each dimension; positive Lyapunov exponent is an important indicator of a chaotic system. In this study, the greatest Lyapunov exponent was calculated on the basis of proposed Rosenstein method. By having the value of optimal embedding dimension, this exponent is calculated. In the case of no optimal embedding dimension, this value is predicted. At the daily and monthly time scales, positive greatest Lyapunov exponent (0.0149 and 0.0373) was obtained.
Hurst exponent is based on his studies on river data obtained from different time periods. By using this exponent, the presence of non-periodic variations in river water flow could be realized. Hurst showed that if the exponent is equal to 0.5, it indicates an independent and completely random process; if 1> H> 0.5, it implies a continuous time series with very long memory; finally, if 0.5 > H> 0, it indicates that the process is not continuous. Based on the obtained results at daily and monthly scales, this exponent was 0.7556 and 0.7773, respectively, for Dez River discharge, and both confirm that the river behavior is chaotic at these two time scales. Besides, the Hurst exponent in a random process may be obtained as positive. Therefore, this river’s discharge can be predicted at daily and monthly scale. Moreover, at seasonal time scale, due to lack of correlation dimension, the flow behavior was shown to be random.